Markov Chains Applications

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Presentation transcript:

Markov Chains Applications

Brand Switching 100 customers currently using brand A 84 will stay with A, 9 will switch to B, 7 will switch to C 100 customers currently using brand B 78 will stay with B, 14 will switch to A, 8 will switch to C 100 customers currently using brand C 90 will stay with c, 4 will switch to A, 6 will switch to B

Brand Switching States Time Period Transition Probabilities Brands of product Time Period Interval between purchases Transition Probabilities A B C P = 84 09 07 14 78 08 04 06 90 .

Brand Switching If the current market share is given by: What is the market share after one time period ? What is long term market share ? P ( ) . 39 32 29 =

Brand Switching Assumptions Markovian Property Brand switching might take into consideration more than just the past brand; e.g. customer dissatisfaction with brand A led to B, marketing lead in led to C Stationarity Property Marketing strategies may change transition probabilities over time

Stock Market Analysis Transition Probabilities 1 P = 4 2

Stock Market Analysis Assumptions Markovian Property Stationarity Property

Equipment Replacement States State 1: New Filter State 2: One year old, no repairs State 3: Two years old, no repairs State 4: Repaired once Time Period One year Transition Probabilities Repaired filter will be replaced after one year Filters scrapped after 3 years use

Equipment Replacement Transition Probabilities 1 3 2 4

Equipment Replacement Transition Probabilities 1 3 2 4 P = 7 3 4 6 5 1 .

Equipment Replacement = F H G I K J 7 3 4 6 5 1 . Steady State Probabilities p 1 3 4 2 5 7 6 = + . p 1 2 3 4 352 246 098 304 = .

Equipment Replacement

Equipment Replacement Suppose that the following costs apply New Filters $ 500 Repair Filter $ 150 Scrap filters ($ 50) (scrap salvage) For a pool of 100 filters, on average, what is the expected cost of our repair policy ?

Population Mobility Forest consists of 4 major species Aspen Birch Oak Maple A B O M P = . 05 08 03 84 00 80 20 35 53 10 85 A B O M

Population Mobility States Time period Transition Probabilities State 1: No movement State 2: Movement within region State 3: Movement out of a region State 4: Movements into a region Time period Transition Probabilities

Population Mobility States: Movement at SDSM&T Time period State 1: Student remains in major State 2: Student switches major State 3: Student leaves school (transfer out, matriculates) State 4: Student enters school (transfer in, first year studs.) Time period Semester Transition Probabilities

Population Mobility States: Movement at SDSM&T Time period State 1: Student in IE State 2: Student in other major State 3: Student leaves school (transfer out, matriculates) State 4: Student enters school (transfer in, first year studs.) Time period Semester Transition Probabilities P = 85 05 10 3 4 9 1 20 70 .